How to use the Kanban board to map out deliverables

We currently use a Jira Sprint board for both the development team and the "delivery and support" team.

The problem is that the Sprint project doesn't seem to be the best planning system for support tasks that's basically doing tasks from the backlog and to also plan out client deliverables with a way to have an outlook on what's coming up.

From what I'm reading, it seems a Kanban project is better for a reactive support environment. Is there some kind of tool that would allow me to enter and easily see client deliverables that would likely be entered as Jira tasks?

Any resources that I could read that would help me clarify what's possible (and what might be common practice)?